We study Bloch-Ogus theory and the Gersten conjecture for homology theories with duality satisfying certain properties, in particular for étale cohomology with finite coefficients coprime to the residue characteristic of the base, for smooth and semi-stable schemes in mixed characteristic. We prove the Gersten conjecture in the smooth case and prove a special case in the semi-stable situation. As a corollary of the smooth case, we obtain the surjectivity of the Galois symbol map for arbitrary local rings over an excellent discrete valuation ring.
We determine the Brauer group of the Deligne-Mumford stack $\mathscr{Y}_0(2)$, the moduli space of elliptic curves with a marked $2$-torsion subgroup over bases of arithmetic interest. Antieau and Meier determine the Brauer group for $\mathscr{M}_{1,1}$, the moduli stack of elliptic curves by exploiting the fact it is covered by the Legendre family and using the Hochschild-Serre spectral sequence. Over an algebraically closed field, Shin uses the coarse space map to determine the Brauer group of $\mathscr{M}_{1,1}$. We combine techniques from both papers to determine the Brauer group of $\mathscr{Y}_0(2)$.
Given a log Calabi-Yau surface (Y,D), Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher-genus descendant logarithmic Gromov-Witten invariants of (Y, D). Our result generalises the weak Frobenius structure conjecture for surfaces to the q-refined setting, and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.
We extend the notions of Hochschild and cyclic homology to morphisms from algebraic spaces to algebraic stacks. Using this, we obtain generalizations to log schemes in the sense of Fontaine and Illusie of these homology theories.
It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves.
